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A338038 a(n) is the sum of the primes and exponents in the prime factorization of n, but ignoring 1-exponents.

%I A338038 #19 May 10 2024 04:08:47
%S A338038 0,2,3,4,5,5,7,5,5,7,11,7,13,9,8,6,17,7,19,9,10,13,23,8,7,15,6,11,29,
%T A338038 10,31,7,14,19,12,9,37,21,16,10,41,12,43,15,10,25,47,9,9,9,20,17,53,8,
%U A338038 16,12,22,31,59,12,61,33,12,8,18,16,67,21,26,14,71,10
%N A338038 a(n) is the sum of the primes and exponents in the prime factorization of n, but ignoring 1-exponents.
%C A338038 First differs from A106492 for n=64.
%H A338038 Robert Israel, <a href="/A338038/b338038.txt">Table of n, a(n) for n = 1..10000</a>
%H A338038 Chris Bispels, Muhammet Boran, Steven J. Miller, Eliel Sosis, and Daniel Tsai, <a href="https://arxiv.org/abs/2405.05267">v-Palindromes: An Analogy to the Palindromes</a>, arXiv:2405.05267 [math.HO], 2024.
%H A338038 Daniel Tsai, <a href="https://arxiv.org/abs/2010.03151">A recurring pattern in natural numbers of a certain property</a>, arXiv:2010.03151 [math.NT], 2020.
%H A338038 Daniel Tsai, <a href="http://math.colgate.edu/~integers/v32/v32.mail.html">A recurring pattern in natural numbers of a certain property</a>, Integers (2021) Vol. 21, Article #A32.
%F A338038 a(n) = A008474(n) for powerful numbers (A001694).
%e A338038 For n = 18 = 2*3^2, a(18) = 2 + (3+2) = 7.
%p A338038 f:= proc(n) local t;
%p A338038   add(t[1]+t[2],t=subs(1=0,ifactors(n)[2]));
%p A338038 end proc:
%p A338038 map(f, [$1..100]); # _Robert Israel_, Oct 13 2020
%t A338038 a[1] = 0; a[n_] := Plus @@ First /@ (f = FactorInteger[n]) + Plus @@ Select[Last /@ f, # > 1 &]; Array[a, 100] (* _Amiram Eldar_, Oct 08 2020 *)
%o A338038 (PARI) a(n) = my(f=factor(n)); vecsum(f[,1]) + sum(k=1, #f~, if (f[k,2]!=1, f[k,2]));
%Y A338038 Cf. A008474, A001694, A106492, A338039.
%K A338038 nonn
%O A338038 1,2
%A A338038 _Michel Marcus_, Oct 08 2020